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Theorem nlimsucg 7084
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
nlimsucg (𝐴𝑉 → ¬ Lim suc 𝐴)

Proof of Theorem nlimsucg
StepHypRef Expression
1 limord 5822 . . . 4 (Lim suc 𝐴 → Ord suc 𝐴)
2 ordsuc 7056 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
31, 2sylibr 224 . . 3 (Lim suc 𝐴 → Ord 𝐴)
4 limuni 5823 . . 3 (Lim suc 𝐴 → suc 𝐴 = suc 𝐴)
5 ordunisuc 7074 . . . . 5 (Ord 𝐴 suc 𝐴 = 𝐴)
65eqeq2d 2661 . . . 4 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 ↔ suc 𝐴 = 𝐴))
7 ordirr 5779 . . . . . 6 (Ord 𝐴 → ¬ 𝐴𝐴)
8 eleq2 2719 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴𝐴𝐴))
98notbid 307 . . . . . 6 (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴𝐴))
107, 9syl5ibrcom 237 . . . . 5 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴))
11 sucidg 5841 . . . . . 6 (𝐴𝑉𝐴 ∈ suc 𝐴)
1211con3i 150 . . . . 5 𝐴 ∈ suc 𝐴 → ¬ 𝐴𝑉)
1310, 12syl6 35 . . . 4 (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴𝑉))
146, 13sylbid 230 . . 3 (Ord 𝐴 → (suc 𝐴 = suc 𝐴 → ¬ 𝐴𝑉))
153, 4, 14sylc 65 . 2 (Lim suc 𝐴 → ¬ 𝐴𝑉)
1615con2i 134 1 (𝐴𝑉 → ¬ Lim suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1523  wcel 2030   cuni 4468  Ord word 5760  Lim wlim 5762  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767
This theorem is referenced by:  tz7.44-2  7548  rankxpsuc  8783  dfrdg2  31825  dfrdg4  32183
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